The present invention relates generally to chromatographic techniques such as high speed liquid chromatography and gas chromatography, and more particularly to methods and apparatuses for analyzing a multichannel chromatogram which is detected by a diode array detector or the like.
A large number of approaches have been proposed for separating or resolving overlapping peaks on a multichannel chromatogram. The multichannel chromatogram refers to a type of a chromatogram having three-dimensional information composed of an absorbance component, a time component, and a wavelength component. Representatives of such approaches include factor analysis and principal component analysis which are techniques of multivariate analysis (see, for example, U.S. Pat. No. 4,807,148 issued to J. K. Strasters et al.; H. R. Keller et at, Journal of Liquid Chromatography, No. 12, pp. 3-22 (1988); J. Craig Hamilton et al., Chemometrics and Intelligent Laboratory Systems, No. 12, pp. 209-224 (1992); Journal of Chemometrics, No. 4, pp. 1-13 (1990), and so on).
A non-linear least-squares analysis has also been proposed as another approach for analyzing a multichannel chromatogram (see, for example, Itoh et al, Abstracts of Seventh Liquid Chromatography Conference, pp. 5 (1986); Itoh et al, Abstracts of 22th Applied Spectrometry Tokyo Conference, pp. 141 (1987); and Itoh et at, Abstracts of 19th HPLC Research Conversation, pp. 30 (1988)). This approach has been developed from a method for fitting overlapping peaks on a multichannel chromatogram obtained from GS/MS using the Gaussian (normal distribution function), which was proposed by F. J. Knorr, by substituting EMG (Exponentially Modified Gaussian) as a more realistic model function for the former Gaussian. This approach will be explained here in detail since it will facilitate the understanding of later descriptions.
Multichannel chromatogram data gathered from a diode array detector is given by the following matrix Dij (1): EQU D=R.multidot.A.multidot.S=R.multidot.X (1)
where i represents a time index, j a wavelength index, k a component index, Dij the absorbance, Rik a normalized retention waveform (chromatogram magnitude), Akk a quantitative factor relative to the concentration of a k-component, Skj a normalized spectral intensity, and Xkj a quantitative spectral intensity multiplied with the factor Akk relative to the concentration of a k-component. It should be noted that the wavelength index j is used as an index of the m/z value when a mass spectrometer is used.
Here, a trial matrix R'ik is introduced in place of the chromatogram intensity matrix Rik. From the equation (1), a trial spectral intensity matrix X'jk can be computed from the trial matrix R'kl and the matrix Dij containing the measured data. Consequently, a trial data matrix D'ij can be obtained as expressed by the following equation (2): EQU D'=R'.multidot.(R'.sup.T .multidot.R').sup.-1 .multidot.R'.multidot..sup.T D(2)
where T in R'.sup.T means a transpose matrix, and -1 in (R'.sup.T .multidot.R').sup.-1 means that the associated matrix is an inverse matrix.
The least-squares method determines parameters so as to minimize Dij-D'ij and provides the best-fit R'ik for each component. In this way, the respective component matrices R'ik and X'ik can be separated or resolved from the matrix Dij including data on overlapping peaks.
This approach utilizes EMG which can also represent a asymmetric tailing peak as the R'ik matrix (the following equation (3)). EQU R'.sub.ik =1(.tau..sub.k .sigma..sub.k (2.pi.).sup.1/2.intg..sub.o.sup.t1 exp[-(t.sub.i -t.sub.RK -t').sup.2 ]/2.sigma..sub.k.sup.2 -t'/.tau..sub.k)]dt' (3)
where t.sub.RK is retention time for a k-component, .sigma..sub.K standard deviation for the k-component, and t.sub.K the time constant for the k-component.
JP-A-60-2447 describes a chromatographic quantitative analyzing method and apparatus which detect temporal changes in absorbance at multiple wavelengths as three-dimensional information to perform quantitative analysis. The chromatographic quantitative analyzing method and apparatus employ functions f1(.lambda.) and f2(.lambda.) which represent two previously measured two-dimensional standard spectra. Then, an equation is obtained for a two-dimensional synthesized component function fs(.lambda.) of a measured sample having over-lapping peaks using the above-mentioned functions f1(.lambda.) and f2(.lambda.). The overlapping peaks are resolved by the obtained equation.
Other approaches utilizing deconvolution have also been proposed as a method for analyzing an ordinary chromatogram, i.e., a chromatogram having two-dimensional information (composed of an absorbance component and a time component). See, for example, U.S. Pat. No. 4,941,101 issued to Paul Benjamine Crilly; Paul Benjamine Crilly, IEEE Transaction on Instrumentation and Measurement, No. 40, pp. 558-562 (1991); and Journal of Chemometrics, No. 5, pp. 85-95 (1991).
Here, the convolution is defined. Original data is detected by a detector, and dispersed by an inherent device function h(t) (dispersion function(instrument function)) which represents the detection characteristics of the detector. The deconvolution is determined to be the processing for removing a dispersion portion of the data by the device function h(t) from the dispersion data. An equation defining the deconvolution is given in the following equation (4): ##EQU1## where D(t) is a detected waveform, d(t) an original waveform, and h(t) a dispersion function.
For the convolution applied to analysis on a chromatogram having two-dimensional information, several approaches have been proposed for promoting the convergence. Specifically, these approaches have been proposed principally relying on iteration methods, and include the Gaussian elimination which performs an inverse matrix operation, as well as Jacobi's method, Gauss-Seidel's method, Fourier Transform method, Van Cittert's method, Constrained Iterative method, Jansson's method, Gold's ratio method, and so on. For details of these methods, see "Waveform Data Processing for Scientific Measurements", edited by Shigeo Minami, published by CQ Editorial, pp. 122-139 (1986); and P. A. Jansson, "Deconvolution with Applications in Spectroscopy, New York, Academic (1984).
Also, a method based on factor analysis for separating or resolving overlapping peaks on a multichannel chromatogram is introduced in detail, for example, by Edmund R. Malinowski, "Factor Analysis in Chemistry", John Wiley & Sons, Inc. (1991).
This factor analysis based method is multivariate analysis, and its basic thinking is that a data matrix D is modeled as a product of a spectral matrix X and an elution profile matrix Y, as represented by the following equations (5) and (6). It should be noted however that the equation (6) defines that each component k is normalized to a peak area of one. However, since the data matrix D cannot be uniformly resolved from a mathematical point of view, several rational constraints are provided to solve the problem. EQU D=XY (5) ##EQU2## where Dij: Signal Magnitude; Xik: Spectral Intensity;
Ykj: Elution Profile; PA1 i: Channel Index; PA1 k: Component Index; PA1 j; Time Index.
Generally, an eigenvalue problem is solved for the data matrix D, the number n of components is determined by principal component analysis, an abstract elution profile matrix V having a characteristic vector as its element is transformed by a matrix T, and thus a physically meaningful elution profile matrix Y is obtained. As the matrix Y is determined, the spectral matrix X can be computed from the data matrix D by the following equations (7)-(15).
More specifically, solving the eigenvalue problem, the data matrix D is represented by a product of an abstract spectral matrix U and the abstract elution profile matrix V as given by the following equation (7): EQU D=UV (7)
Here, a characteristic vector of a product Z=D.sup.T D, i.e., a product of a transposed matrix D.sup.T of the matrix D with the matrix D is Vk. This vector Vk is a k'th row vector of the matrix V given by the following equation (8): ##EQU3##
Also, the relationship between a characteristic value .zeta.k, the characteristic vector Vk, and the matrix Z is given by the following equation (9): EQU ZVk=.zeta.kVk (9)
Further, the vector Vk is such that its sum of squares is normalized to one as given by the following equation (10): ##EQU4##
The matrix V can be transformed to the matrix Y by a transformation matrix T as shown in the following equation (11): EQU Y=TV (11)
where the matrix T is an nxn matrix.
The matrix T serves to perform an oblique rotation and a transformation for transforming the area of each raw vector in the matrix Y to one.
The matrix Y is represented by the following equation (12), where any component Yk of the matrix Y satisfies the following equation (13): ##EQU5##
Also, the matrix X is obtained from the matrices D and Y as shown in the following equation (14): EQU X=DY.sup.T (YY.sup.T).sup.-1 ( 14)
The matrices X, Y, U, and V have the relationship represented by the following equation (15): EQU D=XY=(UT.sup.-1)(TV) (15)
The transformation matrix T, however, cannot be easily determined. Therefore, approaches as follows have been proposed for determining the transformation matrix T.
1. Methods with Known Spectrum:
1-1. TTFA (Target Transformation Factor Analysis): A method for determining a transformation matrix T such that a known spectral waveform is obtained from an abstract spectral matrix.
1-2. RAFA (Rank Annihilation Factor Analysis ): A method for obtaining a known data matrix from standard forms of respective components and subtracting the components one by one from the obtained data matrix.
1-3. GRATA (Generalized Rank Annihilation Factor Analysis):
While RAFA (1-2) requires a data matrix having columns with one component, this approach enables curve resolution to be performed using a data matrix obtained from a standard mixed sample.
2. Modeling Methods with Unknown Spectrum:
2-1. Gaussian Non-linear Least-Squares Method: A method for performing a modeling on assumption that an elution profile is Gaussian, and the model is fit by the non-linear least-squares method.
2-2. Non-linear Least-Squares Method Using the Aforementioned EMG (Exponentially Modified Gaussian): A method identical to the above method 2-1 except for employing EMG which can represent, asymmetric peaks in place of Gaussian.
3. Self Modeling Method with Unknown Spectrum:
3-1. ITTFA (Iterative Target Transformation Factor Analysis): A method which initially introduces a test vector having a pulsatile elution profile and gradually adjusts it to approach to a true elution profile.
3-2. EFA (Evolving Factor Analysis): A method which plots changes in the characteristic value along the time axis to find a stable region of the characteristic value. This stable region is called a window. An approach which fixes the outside of the stable region to zero to determine an elution profile for each component is particularly called WFA (Window Factor Analysis). For details of WFA, see E. R. Malinowski, J, Chemometrics, No. 6, pp. 29-40 (1992); and H. R. Keller et al., Anal. Chim, Acta., No. 246, pp. 379-390 (1991).
3-3. RAEFA (Rank Annihilation by Evolving Factor Analysis):
A method which iterates the peak resolution performed by EFA (3-2) while subtracting components one by one from a data matrix.
3-4. RAFA Using Information Entropy as Index: A method which ranks down a matrix for each component, on the basis of minimum information entropy when obtaining an elution profile (see I. Sakura et al., J. Chromatogr, No. 506, pp. 223-243 (1990)).